Also,
Generalizing from these examples, we have
24 # 23 = 24 +^3 = 27 and 62 # 63 = 62 +^3 = 65.
= 65.
= 6 # 6 # 6 # 6 # 6
62 # 63 = 16 # 6216 # 6 # 62
5.1 The Product Rule and Power Rules for Exponents
This suggests the product rule for exponents.
Product Rule for ExponentsFor any positive integers mand n,
(Keep the same base and add the exponents.)
Example: 62 # 65 = 62 +^5 = 67
am#anamn.
Using the Product RuleUse the product rule for exponents to find each product if possible.
(a) 63 # 65 = 63 +^5 = 68 (b) 1 - 4271 - 422 = 1 - 427 +^2 = 1 - 429
EXAMPLE 3
CAUTION Do not multiply the bases when using the product rule. Keep the
same base and add the exponents.For example,
62 # 65 = 67 , not 367.
NOW TRY
EXERCISE 3
Use the product rule for
exponents to find each product
if possible.
(a)
(b)
(c)
(d)
(e) 32 + 33
24 # 53
12 x^3214 x^72y^2 #y#y^5
1 - 5221 - 524
NOW TRY ANSWERS
- (a) (b) (c)
(d)The product rule does not
apply; 2000 (e)The product
rule does not apply; 36
(-5)^6 y^88 x^10Keep the same base.(c) (d)
(e)
The product rule does not apply, since the bases are different.
23 # 32 = 8 # 9 = 72 Evaluate and Then multiply. 23 32.
23 # 32
x^2 #x=x^2 #x^1 =x^2 +^1 =x^3 m^4 m^3 m^5 = m^4 +^3 +^5 = m^12
Think: 2^3 = 2 # 2 # 2 Think: 3^2 = 3 # 3
(f )
The product rule does not apply, since this is a sum,not a product.
23 + 24 = 8 + 16 = 24 Evaluate and Then add. 23 24.
23 + 24
(g)
Commutative and associative properties
Multiply; product rule= 6 x^10 Add. NOW TRY
= 6 x^3 +^7
= 12 # 32 # 1 x^3 #x^72
12 x^3213 x^722 x^3 means and means 3^2 #x^33 x^7 #x^7.
CAUTION Be sure that you understand the difference between addingand
multiplyingexponential expressions. For example, consider the following.
18 x^3215 x^32 = 18 # 52 x^3 +^3 = 40 x^6